Date: Apr 6, 2013 12:01 AM
Author: namducnguyen
Subject: Re: Matheology § 224
On 05/04/2013 9:38 PM, Nam Nguyen wrote:> On 05/04/2013 9:30 PM, Nam Nguyen wrote:>> On 05/04/2013 8:51 PM, Virgil wrote:>>> In article <nlL7t.371032$O02.109836@newsfe18.iad>,>>> Nam Nguyen <namducnguyen@shaw.ca> wrote:>>>>>>> On 05/04/2013 6:11 PM, Virgil wrote:>>>>> In article <_pJ7t.186674$rk7.149719@newsfe05.iad>,>>>>> Nam Nguyen <namducnguyen@shaw.ca> wrote:>>>>>>>>>>> In so far as a _perceived_ language structure would enable>>>>>> us to interpret the concept of the natural numbers, such>>>>>> a perception is a theology; in it, there are 2 offshoot>>>>>> theologies which we'll _forever_ (i.e. even in principle of>>>>>> logic) struggle to choose for acceptance:>>>>>>>>>>>> - cGC being true>>>>>> - ~cGC being true.>>>>>>>>>> According to Wikipedia>>>>> CGC can be an abbreviation for:>>>>> ¤ Chen Guangcheng a civil rights activist in the People's>>>>> Republic>>>>> of China who drew international attention to human rights issues in>>>>> rural areas>>>>> ¤ Canadian Grenadier Guards>>>>> ¤ Cambridge Gliding Centre>>>>> ¤ Canada Games Company>>>>> ¤ The Capital Group Companies, an investment management>>>>> organization>>>>> ¤ the Canine Good Citizen certification>>>>> ¤ Cerebellar granule cell>>>>> ¤ Certified general contractor, a type of unlimited>>>>> contractor in>>>>> Florida, USA as opposed to registered (limited)>>>>> ¤ Board-Certified Genetic Counselor>>>>> ¤ United States Coast Guard Cutter>>>>> ¤ Color Glass Condensate>>>>> ¤ Comics Guaranty LLC, a grading service for the comic book>>>>> collecting industry>>>>> ¤ Conspicuous Gallantry Cross>>>>> ¤ Constrained geometry complex>>>>> ¤ Career Guidance Council, is a not-to-profit organization>>>>> ¤ Consumer generated content, also known as Consumer generated>>>>> media>>>>> ¤ Co-operative Grocer Chain Japan, known as CGC Japan>>>>>>>>>>>> Sure. Here cGc means the FOL formula written in L(PA) that would stand>>>> as:>>>>>>>> cGC <-> "There are infinitely many counter examples of the Goldbach>>>> Conjecture".>>>>>> Then you presume that the Goldbach conjecture will never be settled?>>> It has not been around as long as the FLT, which finally was settled in>>> the affirmative.>>>> Then you don't seem to understand the nature of cGC, depending on the>> formulation of the Conjecture but being a _different_ formula.>>>> For GC (the Goldbach conjecture), there naturally are 2 cases:>>>> Case 1 - ~GC is true: we found _one specific even natural_ > 4 that>> isn't a sum of two primes.>>>> But that of course has no bearing on either cGC or ~cGC!>>>> So you can't setttle cGC or ~cGC on the account that ~GC>> is true. And ~GC can still be settled as true!>>>> Case 2: GC is true in the naturals as the standard structure for L(PA),>> and it's said NEG(PA |- GC) and NEG(PA |- ~GC).>>>> But if GC is undecidable in PA, there's no proof left in FOL but>> _structure theoretically verifying_ the truth value of GC in>> this structure.>>>> But how would you _verify_ GC be true in this structure?>>>> So, what you have left is just a _pure unverified intuition_>> which is nothing more or less than a mathematical (theology->> like) _belief_ : _no structure theoretical proof_ !>>>> In summary, only in Case 1 could you settle GC, but _in both cases_>> you still can _never_ settle cGC and ~cGC.>> Of course this is just a summary, a bird eye view.>> You can prove that it's impossible to structure theoretically> verify the truth value of either cGC or ~cGC, using some logic> anti-induction principles.>> The problem is that you, we, have been so "trained" on IP (Induction> Principle) that we _don't even suspect_ IP has some loopholes.>> It does!

Seriously, we should begin to abandon the idea that whatever is trueor false in the naturals can be structure theoretically proven,verified.

If we don't, we'd be in _no_ better position than where Hilbertwas with his All-mighty-formal-system, proving all arithmetictrue formulas.

We'd be simply change the name "All-mighty-formal-system"to "All-mighty-language-structure". But it's still an Incompleteness(of the 2nd kind) that we'd encounter: the Incompleteness of languagestructure interpretation of the abstract (non-logical) concept knownas the natural numbers.

We can not as a matter of logic prove the truth value of cGC and ~cGCin the perceived natural numbers, as long as we have the concept ofthe natural numbers.

cGC, ~cGC aren't in the same class of FLT, or even GC.

-- ----------------------------------------------------There is no remainder in the mathematics of infinity.